I'm studying a two dimensional lattice with delta functions at the lattice points. When I diagonalized the hamiltonian I noticed that if I kept the dimension of my matrix less than 5 I would get band gaps at all the BZ boundaries. However, if I increased my matrix to a dimension 5 or higher, some of the bands never got a band gap at certain boundaries despite a huge potential! The picture shows the first 5 energy bands (used dimension 9 matrix because it is more symmetric along the diagonal) for a .5 potential in atomic units.(adsbygoogle = window.adsbygoogle || []).push({});

So I was trying to figure out why this was happening, and I thought that perhaps it's because the roots of 5th order polynomials and higher can't be expressed in terms of radicals. This is a stretch, but maybe the potential-dependent separation between bands at the BZ boundaries is in part a result of having radicals in the solution to the eigenvalues problem. This doesn't happen when using a 1D lattice however, so why would radicals only matter in 2D?

My other guess is that it has something to do with symmetry breaking. Perhaps the symmetry isn't broken at these points for some reason. I wouldn't know how to back this up however. Could anyone tell me why there are no band gaps at some locations for a hamiltonian 5x5 and higher?

Thank you!

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# Question about band gaps in 2D

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